We end this introduction with some remarks concerning local existence of solutions when the **equations** are defined on the whole space. D. Khoshnevisan pointed to us that since for any fixed t > 0, the last term of (1.6) grows like √ log x as x goes to infinity, the solution to (1.6) might blow up instantaneously. That is, any solution of (1.6) can blow up for any t > 0 so that there is no local solution. In the deterministic setting, similar phenomenon arises; see for instance [16] where the exponential reaction-diffusion is studied. Proving such non- existence results is beyond the scope of this paper where the main concern is non-existence of global solution. The above result for instance makes no claim about the existence of a local solution.

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However, comparing with **stochastic** **partial** diﬀerential **equations** with delays, there are only a few results about neutral **stochastic** **partial** diﬀerential **equations**. Precisely, Liu 11 considered a linear neutral **stochastic** diﬀerential **equations** with constant delays and some stability properties of the mild solutions in a similar way as Datko 25 in the deterministic case. Caraballo et al., in 3, have studied the almost sure exponential stability and ultimate boundedness of the solutions to a class of neutral **stochastic** semilinear **partial** delay diﬀerential **equations**; Mahmudov, in 14, has discussed the existence and uniqueness for mild solution of neutral **stochastic** diﬀerential **equations** by constructing a new iterative scheme under the non-Lipschitz conditions.

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solving SDDEs in various fields. It can also be shown in this research, the SRK methods are easy to implement compare to the approximation methods obtained from the truncating **stochastic** Taylor expansion. In this way the computation of high–order **partial** derivatives can be avoided. Moreover, the generalization of convergence proof when the drift and diffusion functions are Taylor expansion is hoped can facilitate mathematicians to explore this area more widely.

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BOUNDARY VALUE PROBLEHS FOR STOCHASTIC DIFFERENTIAL EQUATIONS Thesis by Thomas 1 Tilliam HacDm rell In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institu[.]

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The second application would be to provide a numerical scheme for approximating the solution of a forward backward **stochastic** diﬀerential equation (FBSDE) with ran- dom coeﬃcients. When the coeﬃcients are deterministic the FBSDE (i.e. Markovian case) can be solved by using the Four Step Scheme of Ma, Protter and Yong [8], which requires that a deterministic PDE be solved. When the coeﬃcients of the FBSDE are random, it has been shown in Ma and Yong [9, 10, 11] that the Four Step Scheme requires the solution of parabolic and elliptic SPDEs for the ﬁ nite and in ﬁ nite cases, respectively.

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reproduce and check many of the details described by Roberts [2]. We consider a small spatial domain, representing a finite element, and ap- ply **stochastic** centre manifold techniques to derive a one degree of freedom model for the dynamics in the element. The approach auto- matically parametrises the microscale structures induced by spatially varying **stochastic** noise within the element. The crucial aspect of this work is that we explore how many noise processes may interact in nonlinear dynamics.

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Direct tracking of KL expansions. In the popular numerical methods for SPDEs, such as MC, qMC, gPC, and gSC, additional post-processing steps are necessary to get the KL expansions of **stochastic** solutions. Without any additional post-processing steps, DyBO methods explore the inherent low-dimensional structures and give directly the **stochastic** solutions in bi-orthogonal form that essentially tracks the KL expansion of the **stochastic** solutions. The ability to preserve bi- orthogonal forms in DyBO methods has been proved rigorously and verified numerically in various challenging problems, such as **stochastic** Burgers **equations**, and **stochastic** flows in 2D unit square. Reduced-order models and computation reductions. An important benefit associated with DyBO formulations is the significant savings of computational costs both in memory consumptions and computational times. Detailed complexity analysis has been conducted for DyBO-gPC methods for certain classes of time-dependent SPDEs and verified numerically in Chapter 4.2. In practice, we have observed speedup up to 200 times compared to gPC methods in 2D **stochastic** flow problems.

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(β − α) −1/2 (and make similar assumption about the diffusion coefficient B), see **Condition** 2.1 given in the next section, and prove the existence and uniqueness of finite time solutions of the corresponding Cauchy problem. Observe that the singularity allowed here is weaker than in the deterministic case (cf. (1.2)), which is related to the specifics of the Ito integral estimates. As in the deterministic case, the solution will live in the scale X α , α ∈ A . For simplicity, we assume

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with A(t, ω) being a predictable linear operator on H that belongs to L(V ; V ′ ), where (V, H, V ′ ) is a rigged Hilbert space or the called Gelfand’s triple. Now assuming that A(t, ω) is coercive for a.a. (t, ω) ∈ [0, T ] × Ω and F satisﬁes a similar **condition** to the ones given earlier, we shall show that this BSPDE admits a unique solution (Y, Z, N ) of predictable processes taking values in V × L 2 (H) × M 2,c (H) and that Y is a continuous semimartingale. This space

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DOI: 10.4236/am.2019.1011063 878 Applied Mathematics Section 2). 3) Stratonovich’s interpretation has sounder mathematical and phys- ical bases: a) the ordinary rules of calculus are preserved; b) it is invariant with respect to a time reversal and guarantees the **condition** of detailed balance [29]; c) in the case of a dynamical system it respects the law of the energy conservation [32]. 4) The kinetic interpretation is the only that agrees with the law of the thermodynamic [11] [27]. 5) The experiments are in accord with Stratonovich’s interpretation [20] [21]. 6) The FPK equation deriving from Stratonovich’s in- terpretation has one more term in the drift, the so-called spurious drift that does not originate from a physical reason (it is recalled that this term coincides with the Wong-Zakai-Stratonovich corrective term [33] [34], which makes Itô’s solu- tion of the FPK equation equal to Stratonovich’s one). In [16] it is supposed that the spurious drift is caused by the infinitely fast fluctuations of the Gaussian white noise. No attempt is made here to give a meaning to the spurious drift.

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Due to the mathematically complex nature of **stochastic** processes, there are limited cases in which SPDEs can be solved analytically. Moreover, contemporary numerical algorithms are not optimal for deriving an exact form of their solutions either; so, we are compelled to seek an accurate and efficient method to better approximate these **equations**. A newly developing method to approximate the solution of SPDEs involves the use of cellular automata. The utilization of cellular automata allows one to represent the diffusion processes associated with SPDEs in a discretized n-dimensional mesh-based grid, like a dynamical system. The concept of cellular automaton was developed by Neumann and Ulmann in the 1940s. This topic developed throughout the 20th and 21st centuries through the popularization of various models, such as Conway’s Game of Life. In 2002, Wolfram published A New Kind of Science [2] in which he asserts that the use of cellular automaton has a multitude of applications. The work discussed in this paper is a continuation on this foundational work.

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Most of the existing results on **stochastic** stability use a single Lyapunov function, but we shall instead use multiple Lyapunov functions in this paper. We shall establish the sufficient **condition**, in terms of multiple Lyapunov functions, for the asymptotic behaviours of solutions of stochas- tic **differential** delay **equations**. Moreover, from them follow many effective criteria on **stochastic** asymptotic stability, which enable us to construct the Lyapunov functions much more easily in applications. In particular, the well-known classical theorem on **stochastic** asymptotic stability is a special case of our more general results. These show clearly the power of our new results. Two examples are also given for illustration.

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In this section we describe a novel class of bridge constructs that require no tuning parameters, are simple to implement (even when only a subset of components are observed with Gaussian noise) and can account for nonlinear dynamics driven by the drift. In addition, we discuss the recently pro- posed bridging strategy of Schauer et al. (2016) and describe an implementation method in the case of **partial** observation with additive Gaussian measurement error.

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other types of transforms to study similar **equations**. Indeed the transfor- mation introduced by Zvonkin in [27], when the drift is a function, is also stated in the multidimensional case. In a series of papers the first named author and coauthors (see for instance [9]), have efficiently made use of a (multidimensional) Zvonkin type transform for the study of an SDE with measurable not necessarily bounded drift, which however is still a function. Zvonkin transform consisted there to transform a solution X of (2) (which makes sense being a classical SDE) through a solution ϕ : [0, T ] × R d → R d of a PDE which is close to the associated Kolmogorov equation (3) with some suitable final **condition**. The resulting process Y with Y t = ϕ(t, X t )

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determined by a quasi-linear **partial** **differential** equation of parabolic type. Recently, Bouchard and Touzi [4] propose a Monte-Carlo approach which may be more suitable for high-dimensional problems. Again in this forward-backward setting, if the genera- tor f has a quadratic growth in Z, a numerical approximation is developed by Imkeller and Dos Reis [20] in which a truncation procedure is applied.

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chatacterised by a single representative agent maximising expected utility for wealth at terminal date H, they demonstrate, by using binormial tree argument in the discrete time setting and Taylor expansion, that a necessary **condition** for economic equilibria is that the ratio α follows the following Burgers equation

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This definition of the class Car (Ã × [a, b], μ) concerning the map g(x, s)(𝜂, 𝜉) is closely related to the class C (Ã × [a, b], W) in [1]. Indeed, if μ is the Lebesgue measure W (t) = t on [a, b], then they are the same except that (2.3) and (2.4) here are required to hold everywhere instead of μ - almost everywhere. In the definition of the class Car (Ã × [a, b], μ), (2.5) from Definition 2.1 of the class C (Ã × [a, b], μ, W) is replaced by (2.10). The **condition** expressed by (2.5) requires that the continuity from (2.10) has a given modulus W. It is obvious that C (Ã × [a, b], μ, W) Car (Ã × [a, b], μ).

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motivated by [–], we study the exponential stability of impulsive **stochastic** neutral **partial** diﬀerential **equations** with memory by establishing a new integral inequality. The results obtained here generalize the main results from [, , ] to cover a class of more general impulsive **stochastic** neutral systems.

In recent years, fractional Lévy processes are getting popular since they are more ﬂexible in modeling the distributions of noises than fractional Brownian motions (FBMs). More pre- cisely, they can capture large jumps and model high variability in the real systems appear- ing in ﬁnance, telecommunications, and so on. Meanwhile, they can also capture the long memory eﬀect in a similar way as FBMs do (see [1, 6–9, 12, 13], etc.). Currently, more and more researchers have been attracted to the studies of fractional Lévy processes, **stochastic** calculus for fractional Lévy processes, and **stochastic** diﬀerential **equations** driven by these processes. In [14], the authors deﬁned a **stochastic** integral for a class of deterministic inte- grands with respect to real-valued fractional Lévy processes. In [13], we deﬁned a stochas- tic integral for a class of real deterministic functions and deterministic operator-valued processes with respect to fractional Lévy processes on Gel’fand triple. In [1], by using S- transform the authors investigated the Skorokhod integral for fractional Lévy processes whose underlying Lévy processes have ﬁnite moments of any order by avoiding Malliavin calculus and white noise analysis. In [11], a white noise theory for fractional Lévy process was developed by considering it as a generalized functional of the sample path of Lévy process and solving several kinds of **stochastic** ordinary diﬀerential **equations** driven by fractional Lévy noises.

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Abstract We consider the numerical solution, by finite differences, of second-order-in-time **stochastic** par- tial **differential** **equations** (SPDEs) in one space dimension. New timestepping methods are introduced by generalising recently-introduced methods for second-order-in-time **stochastic** **differential** **equations** to multi- dimensional systems. These **stochastic** methods, based on leapfrog and Runge-Kutta methods, are designed to give good approximations to the stationary variances and the correlations in the position and veloc- ity variables. In particular, we introduce the reverse leapfrog method and **stochastic** Runge-Kutta Leapfrog methods, analyse their performance applied to linear SPDEs and perform numerical experiments to examine their accuracy applied to a type of nonlinear SPDE.

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